processes

Article

Improved Genetic Algorithm Tuning ControllerDesign for Autonomous Hovercraft

Huu Khoa Tran 1,2, Hoang Hai Son 3, Phan Van Duc 4, Tran Thanh Trang 5

and Hoang-Nam Nguyen 6,*1 Industry 4.0 Center, National Taiwan University of Science and Technology, Taipei 10607, Taiwan;

khoa.tran@mail.ntust.edu.tw2 Center for Cyber-Physical System Innovation, National Taiwan University of Science and Technology,

Taipei 10607, Taiwan3 Faculty of Mechanical, Electrical, Electronic and Automotive Engineering, Nguyen Tat Thanh University,

Ho Chi Minh City 700000, Vietnam; hhson@ntt.edu.vn4 Faculty of Automobile Technology, Van Lang University, Ho Chi Minh city 700000, Vietnam;

phanvanduc@vanlanguni.edu.vn5 Faculty of Engineering and Technology, Van Hien University, 665-667-669 Dien Bien Phu,

Ho Chi Minh city 700000, Vietnam; trangtt@vhu.edu.vn6 Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics

Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam* Correspondence: nguyenhoangnam@tdtu.edu.vn

Received: 27 November 2019; Accepted: 26 December 2019; Published: 3 January 2020�����������������

Abstract: By mimicking the biological evolution process, genetic algorithm (GA) methodology hasthe advantages of creating and updating new elite parameters for optimization processes, especiallyin controller design technique. In this paper, a GA improvement that can speed up convergence andsave operation time by neglecting chromosome decoding step is proposed to find the optimizedfuzzy-proportional-integral-derivative (fuzzy-PID) control parameters. Due to minimizing trackingerror of the controller design criterion, the fitness function integral of square error (ISE) was employedto utilize the advantages of the modified GA. The proposed method was then applied to a novelautonomous hovercraft motion model to display the superiority to the standard GA.

Keywords: modified GA; fuzzy-PID control; autonomous hovercraft; ISE criterion

1. Introduction

John Henry Holland, by imitating Darwin’s biological evolution process, proposed the powerfulstochastic global search method genetic algorithm (GA) first in 1975 [1,2]. Both the two reproductionmechanisms of genetic algorithm, including crossover and mutation, are used to find the convergenceof optimal solutions. These values show an important effect on both behavior and performance.Therefore, GA instructions for choosing appropriate values are introduced by many researchers.In 1986, Grefenstette et al. proposed a method for optimizing the control gains for the geneticalgorithm [3]. Then, in 1994, Srinivas and Patnaik demonstrated the adaptive probabilities of crossoverand mutation [4]. In 1997, the bounded difficulty problems were considered by Harik [5]. Later in 2004,Zlochin et al. suggested model-based search for implementing combinatorial optimization [6]. In 2007,Zhang et al. adaptively adjusted crossover and mutation probabilities by utilizing a clustering-basedtechnique [7]. Preceding this paper, GA and its innovations have been successfully deployed in awide range of non-trivial complicated real-world issues, from optimization of flight control laws [8] toaerodynamic optimization problems [9]; from small wind turbine design [10] to path planning of a

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space-based of a manipulator system [11]; and from modeling collinear data [12] to ship navigation incollision situations [13].

The genetic algorithm optimization offers a powerful methodology for solving single tomultivariable problems. Each GA represents a problem solution encoded by the form of binarystrings or chromosomes. Its fitness is employed to measure how good of one solution by increasing thebest bit patterns. Hence, the maxima/minima of fitness function value are then optimized during theGA course.

Nevertheless, the standard GA (denoted shortly as sGA) still has some drawbacks; for example, lowconvergence speed and premature convergence because of requiring hundreds of updated generations.Hence, we proposed a modified genetic algorithm (denoted shortly as mGA) to improve the optimalprocess. This modified algorithm, just in short operating generations [1,2,14], can improve the globalsearch efficiency and increase the convergence speed of the optimal control design. Next, the proposedcontrol design was a fuzzy-proportional-integral-derivative (F-PID) control [15–25] that comprisesthe fuzzy logic control (FLC) and the common PID controller. The first, fuzzy term is employed toincrease the stability and the robustness of the controller design by tuning the membership functionsand by selecting suitable methods of fuzzification and defuzzification [15–19,24,25]. The second,PID control term is separated in two small sub-terms: the PD is employed to maintain the systemstability while the term I is utilized to eliminating the steady-state error of the controlled systemresponse [20–23]. Based on the calculation of criterion error of the control system, the “integral ofsquare error” (ISE) [20–22] was chosen as a fitness function to show the controller performance index.

A hovercraft, also known as an air cushion vehicle (ACV), moves smoothly on any surface [26–31]from the ground-land to the mud, water, sand, and even on ice. Because a hovercraft is very active andagile, its models was chosen for the control implementation.

To summarize, the modified GA is proposed to take advantages of the fuzzy-PID controller designin shorter operation period. Thus, the dynamic of Hovercraft models could be mobility in high stabilitywith fast response and less error.

2. Hovercraft Prototype Model

A hovercraft, which is known as an underactuated system and named an air cushion vehicle(ACV), has rotors and a cushion, where inside air pressure enables floating and smooth movement onany surface [26–31], such as land, mud, water, sand, and even on ice. The hovercraft is very activeand agile; hence, it is applied widely in the coastguard, army, rescue operations, civil engineering, etc.The hovercraft is mounted with a single tilt servomotor on the fin tail. As shown in Figure 1, the rotorduct fan of the hovercraft is settled along the y-axis, while the propeller is attached along the z-axis.Firstly, the lift propeller provides the internal cushion pressure to lift up in a long operation period.Next, the forward moving is created by the rear rotor duct-fan. Finally, the turning typically is operatedby directing the thrust airflow through rotor duct fan, which is steered by a tilt servomotor placedat the rear. The subsequently generated momentum is used to maneuver the craft. Although manymodern technologies are utilized, the hovercraft still requires an advanced maneuvering system toachieve optimized performance.

The hovercraft dynamic models were derived in [28–31] using right-hand convention coordinatesystems. The positive x-axis covers the lateral factors, namely, sway motion or surge position, whilethe y-axis is the direction along the hovercraft body, covering surge motion or sway position; andthe positive z-axis defines the downwards direction. The hovercraft’s kinematics can be expressed asEquation (1):

.x = ucosψ− vsinψ.y = vcosψ+ usinψ

.ψ = r

(1)

where u ∈ R and v ∈ R represent linear velocities in surge direction and sway direction, respectively.The angular velocity is represented as r ∈ R. Using Equation (1), we can derive the kinetic energy T

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and potential energy V to define the Lagrange L = T − V by applying Euler–Lagrange formulation asin following equation:

M(q).q + C

(q,

.q)q =

Fτ0

(2)

where τ is the torque in yaw rotation and F is the force acting along surge direction.

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Figure 1. The hovercraft configuration model.

The hovercraft dynamic models were derived in [28–31] using right-hand convention coordinate systems. The positive x-axis covers the lateral factors, namely, sway motion or surge position, while the y-axis is the direction along the hovercraft body, covering surge motion or sway position; and the positive z-axis defines the downwards direction. The hovercraft’s kinematics can be expressed as Equation (1): 𝑥 𝑢𝑐𝑜𝑠𝜓 𝑣𝑠𝑖𝑛𝜓𝑦 𝑣𝑐𝑜𝑠𝜓 𝑢𝑠𝑖𝑛𝜓𝜓 𝑟 (1)

where 𝑢 ∈ 𝑅 and 𝑣 ∈ 𝑅 represent linear velocities in surge direction and sway direction, respectively. The angular velocity is represented as 𝑟 ∈ 𝑅. Using Equation (1), we can derive the kinetic energy T and potential energy V to define the Lagrange L = T − V by applying Euler–Lagrange formulation as in following equation:

𝑀 𝑞 𝑞 𝐶 𝑞, 𝑞 𝑞 𝐹𝜏0 (2)

where τ is the torque in yaw rotation and F is the force acting along surge direction.

3. Modified GA Optimal Controller Gains

3.1. The Controller Design

Fuzzy logic concepts are extremely modest but powerful and effective for applications in the control of various machines. The fuzzy control takes advantages in stability and robustness since its aptitude is deal with the nonlinear and uncertain systems. Fuzzy control rules are designed based on the center of area method (COA) for defuzzification and Mandani’s MIN–MAX inference engine type. It includes the tracking error e(t) and the differential tracking error de(t) as the inputs. The fuzzy control output to PID control is defined as eFuzzy(t). Seven partitions of the fuzzy rule-table control, which exploit the triangular membership functions (as shown in Figure 2), are noted in Table 1, which include negative small (NS), negative medium (NM), negative big (NB), positive big (PB), positive medium (PM), positive small (PS), and zero (Z).

Tilt Rotor Propeller Rotor Duct Fan

z

y

x

ϕθ

ψ

Figure 1. The hovercraft configuration model.

3. Modified GA Optimal Controller Gains

3.1. The Controller Design

Fuzzy logic concepts are extremely modest but powerful and effective for applications in thecontrol of various machines. The fuzzy control takes advantages in stability and robustness since itsaptitude is deal with the nonlinear and uncertain systems. Fuzzy control rules are designed basedon the center of area method (COA) for defuzzification and Mandani’s MIN–MAX inference enginetype. It includes the tracking error e(t) and the differential tracking error de(t) as the inputs. The fuzzycontrol output to PID control is defined as eFuzzy(t). Seven partitions of the fuzzy rule-table control,which exploit the triangular membership functions (as shown in Figure 2), are noted in Table 1, whichinclude negative small (NS), negative medium (NM), negative big (NB), positive big (PB), positivemedium (PM), positive small (PS), and zero (Z).

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Figure 2. The fuzzification membership function.

Table 1. Fuzzy rule-table.

eFuzzy(t) e(t)

NB NM NS Z PS PM PB

de(t)

PB Z PS PM PB PB PB PB PM NS Z PS PM PB PB PB PS NM NS Z PS PM PB PB Z NB NM NS Z PS PM PB

NS NB NB NM NS Z PS PM NM NB NB NB NM NS Z PS NB NB NB NB NB NM NS Z

PID controllers are designed to satisfy dynamic response, and reduce and/or eliminate error of physical empirical systems. Thus, the fuzzy-PID controller [24,25] output u to the system is proposed and expressed on Equation (3) as: 𝑢 𝑢 𝑢 𝑢 𝑘 𝑒 𝑘 𝑒𝑑𝑡 𝑘 𝑑𝑑𝑡 𝑒 𝑡 (3)

where 𝑢 and 𝑢 are the Fuzzy and PID controller output signals. 𝑘 , 𝑘 , 𝑘 are the PID controller proportional, integral and derivative constants.

The fuzzy-PID controls is highly effective, as it is simple, has less overshoot, and is able to eliminate the steady state error, and especially smooth control signal. Its block diagram is demonstrated in Figure 3.

Figure 3. The controller block diagram.

3.2. Improved GA in the Optimization Process

The standard GA’s characteristics [3], which includes three main parameters—mutation rate (Rm), crossover rate (Rc), and population size (N)—were employed to verify the optimized gains of

-1 -0.7 -0.4 0 0.4 0.7 10

0.2

0.4

0.6

0.8

1

1.2

Input: e and de, Output CI

mem

bers

hip

func

tion

NS NM NB ZE PS PM PB

Autonomous Hovercraft

models (X, Y, steering)

Fuzzy PID

Controller

+ -

y e u r

Figure 2. The fuzzification membership function.

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Table 1. Fuzzy rule-table.

eFuzzy(t)e(t)

NB NM NS Z PS PM PB

de(t)

PB Z PS PM PB PB PB PBPM NS Z PS PM PB PB PBPS NM NS Z PS PM PB PBZ NB NM NS Z PS PM PB

NS NB NB NM NS Z PS PMNM NB NB NB NM NS Z PSNB NB NB NB NB NM NS Z

PID controllers are designed to satisfy dynamic response, and reduce and/or eliminate error ofphysical empirical systems. Thus, the fuzzy-PID controller [24,25] output u to the system is proposedand expressed on Equation (3) as:

u = uFuzzy + uPID = uFuzzy + kp × e + kI

∫edt + kD ×

ddt

e(t) (3)

where uFuzzy and uPID are the Fuzzy and PID controller output signals. kP, kI, kD are the PID controllerproportional, integral and derivative constants.

The fuzzy-PID controls is highly effective, as it is simple, has less overshoot, and is able to eliminatethe steady state error, and especially smooth control signal. Its block diagram is demonstrated inFigure 3.

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Figure 2. The fuzzification membership function.

Table 1. Fuzzy rule-table.

eFuzzy(t) e(t)

NB NM NS Z PS PM PB

de(t)

PB Z PS PM PB PB PB PB PM NS Z PS PM PB PB PB PS NM NS Z PS PM PB PB Z NB NM NS Z PS PM PB

NS NB NB NM NS Z PS PM NM NB NB NB NM NS Z PS NB NB NB NB NB NM NS Z

PID controllers are designed to satisfy dynamic response, and reduce and/or eliminate error of physical empirical systems. Thus, the fuzzy-PID controller [24,25] output u to the system is proposed and expressed on Equation (3) as: 𝑢 𝑢 𝑢 𝑢 𝑘 𝑒 𝑘 𝑒𝑑𝑡 𝑘 𝑑𝑑𝑡 𝑒 𝑡 (3)

where 𝑢 and 𝑢 are the Fuzzy and PID controller output signals. 𝑘 , 𝑘 , 𝑘 are the PID controller proportional, integral and derivative constants.

The fuzzy-PID controls is highly effective, as it is simple, has less overshoot, and is able to eliminate the steady state error, and especially smooth control signal. Its block diagram is demonstrated in Figure 3.

Figure 3. The controller block diagram.

3.2. Improved GA in the Optimization Process

The standard GA’s characteristics [3], which includes three main parameters—mutation rate (Rm), crossover rate (Rc), and population size (N)—were employed to verify the optimized gains of

-1 -0.7 -0.4 0 0.4 0.7 10

0.2

0.4

0.6

0.8

1

1.2

Input: e and de, Output CI

mem

bers

hip

func

tion

NS NM NB ZE PS PM PB

Autonomous Hovercraft

models (X, Y, steering)

Fuzzy PID

Controller

+ -

y e u r

Figure 3. The controller block diagram.

3.2. Improved GA in the Optimization Process

The standard GA’s characteristics [3], which includes three main parameters—mutation rate (Rm),crossover rate (Rc), and population size (N)—were employed to verify the optimized gains of theproposed control system. This optimal program usually takes hundreds of generations to update andfind out convergence gains. Therefore, the GA is powerful, but it still has a critical drawback, whichhas been investigated for a solution by many researchers. By checking the GA process, it was foundout that chromosome decode is not compulsory due to the evaluation of the cost function [1,2,14].Hence, we proposed the novel modified GA method where the chromosome decoding step is totallyneglected, as shown in Figure 4. This improvement makes modified GA more effective than theconventional GA in the aspects of storage (less) and convergence speed (naturally higher) [1,2,14].The optimal process of the modified GA to Fuzzy-PID control parameters using the ISE (integral ofsquare errror) fitness function is formulated as ISE =

∫∞

0 e2(t)dt.

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the proposed control system. This optimal program usually takes hundreds of generations to update and find out convergence gains. Therefore, the GA is powerful, but it still has a critical drawback, which has been investigated for a solution by many researchers. By checking the GA process, it was found out that chromosome decode is not compulsory due to the evaluation of the cost function [1,2,14]. Hence, we proposed the novel modified GA method where the chromosome decoding step is totally neglected, as shown in Figure 4. This improvement makes modified GA more effective than the conventional GA in the aspects of storage (less) and convergence speed (naturally higher) [1,2,14]. The optimal process of the modified GA to Fuzzy-PID control parameters using the ISE (integral of square errror) fitness function is formulated as 𝐼𝑆𝐸 𝑒 𝑡 𝑑𝑡.

Figure 4. The proposed fuzzy-PID-mGA (fuzzy-proportional-integral-derivative modified genetic algorithm; the decode chromosome step is neglected).

4. Numerical Simulation Results

The autonomous hovercraft simulation parameters were derived from [28–31] and denoted with the mass m = 2.1 kg and the inertia moment I = 0.000257 kg.m2. The modified GA had the mutation rate (Rm) of 0.08, the crossover rate (Rc) of 0.95, and one hundred individuals (n = 100). We proposed to tune the fuzzy-PID controller gains in only 20 population-generations of the simulation process. The PID gains were chosen arbitrary initialized in range (0,50) while e and de of fuzzy control changed slightly around 1; the control parameters after optimization by the mGA are in Table 2. In comparison with the standard GA, the modified GA operated in shorter generations (just 20 generations) and rapidly updated the convergence speed of fitness function.

Figure 4. The proposed fuzzy-PID-mGA (fuzzy-proportional-integral-derivative modified geneticalgorithm; the decode chromosome step is neglected).

4. Numerical Simulation Results

The autonomous hovercraft simulation parameters were derived from [28–31] and denoted withthe mass m = 2.1 kg and the inertia moment I = 0.000257 kg.m2. The modified GA had the mutationrate (Rm) of 0.08, the crossover rate (Rc) of 0.95, and one hundred individuals (n = 100). We proposedto tune the fuzzy-PID controller gains in only 20 population-generations of the simulation process.The PID gains were chosen arbitrary initialized in range (0,50) while e and de of fuzzy control changedslightly around 1; the control parameters after optimization by the mGA are in Table 2. In comparisonwith the standard GA, the modified GA operated in shorter generations (just 20 generations) andrapidly updated the convergence speed of fitness function.

The hovercraft was tested by moving forward (x-direction); stability was tested when it wasattacked on the side, as it would be by a wave or the wind (y-direction); and steering was by pilotcontrol (z-direction). Hence, the subsystems of the hovercraft are separated in each channel for thestrategy of design the controller. Performance of the proposed controller of the autonomous Hovercraftmotions are shown on surge position on x-axis in Figure 5, sway position on y-axis in Figure 6, andyaw angle on z-axis in Figure 7, respectively. The numerical simulation result, achieved in just after20 generations, clearly proves that the proposed methodology gives a fast response, less error, andzero overshoot. Moreover, the ISE fitness function, as shown in part b of each of Figures 5–7, gives thebetter process of the minimum error. In all three of Figures 5–7, the modified GA error from start tofinish is faster than the standard GA, as shown on Table 3. The 1/Fitness error being larger means thatthe speed of the convergence from start toward finish is faster. All the simulation performance workwas done on the MATLAB/Simulink platform (R2018a, MathWorks, Natick, MA, USA).

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Table 2. The fuzzy-PID-mGA tuning results.

Generation e de KI KP KD Surge position x (5 cm) 20 1 1 41.61 8.75 1.42 Sway position y (5 cm) 20 0.98 1.05 29.3 3.32 0.66

Steering-Yaw angle (5 degree) 20 1.1 1.03 32.8 5.91 3.26

The hovercraft was tested by moving forward (x-direction); stability was tested when it was attacked on the side, as it would be by a wave or the wind (y-direction); and steering was by pilot control (z-direction). Hence, the subsystems of the hovercraft are separated in each channel for the strategy of design the controller. Performance of the proposed controller of the autonomous Hovercraft motions are shown on surge position on x-axis in Figure 5, sway position on y-axis in Figure 6, and yaw angle on z-axis in Figure 7, respectively. The numerical simulation result, achieved in just after 20 generations, clearly proves that the proposed methodology gives a fast response, less error, and zero overshoot. Moreover, the ISE fitness function, as shown in part b of each of Figures 5–7, gives the better process of the minimum error. In all three of Figures 5–7, the modified GA error from start to finish is faster than the standard GA, as shown on Table 3. The 1/Fitness error being larger means that the speed of the convergence from start toward finish is faster. All the simulation performance work was done on the MATLAB/Simulink platform (R2018a, MathWorks, Natick, MA, USA).

Table 3. The error fitness function by generations.

1/Fitness Error i Gene- Ration

sGA mGA i Gene- Ration

sGA mGA

Surge position x (5 cm) 3 0.00137 0.00236 20 0.0018 0.00263 Sway position y (5 cm) 10 0.045 0.08 20 0.045 0.081 Steering-Yaw angle (5 degree) 10 0.00035 0.00067 20 0.00036 0.00095

(top) Processes 2020, 8, x FOR PEER REVIEW 7 of 10

(bottom)

Figure 5. (Top) Surge—position control; (bottom) improved GA (mGA) versus standard GA (sGA) fitness functions.

(top)

(bottom)

Figure 6. (Top) Sway—position control; (bottom) improved GA (mGA) versus standard GA (sGA) fitness functions.

Figure 5. (Top) Surge—position control; (bottom) improved GA (mGA) versus standard GA (sGA)fitness functions.

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(bottom)

Figure 5. (Top) Surge—position control; (bottom) improved GA (mGA) versus standard GA (sGA) fitness functions.

(top)

(bottom)

Figure 6. (Top) Sway—position control; (bottom) improved GA (mGA) versus standard GA (sGA) fitness functions.

Figure 6. (Top) Sway—position control; (bottom) improved GA (mGA) versus standard GA (sGA)fitness functions.

Table 2. The fuzzy-PID-mGA tuning results.

Generation e de KI KP KD

Surge position x (5 cm) 20 1 1 41.61 8.75 1.42Sway position y (5 cm) 20 0.98 1.05 29.3 3.32 0.66

Steering-Yaw angle (5 degree) 20 1.1 1.03 32.8 5.91 3.26

Table 3. The error fitness function by generations.

1/Fitness Error i Gene-Ration sGA mGA i Gene-Ration sGA mGA

Surge position x (5 cm) 3 0.00137 0.00236 20 0.0018 0.00263Sway position y (5 cm) 10 0.045 0.08 20 0.045 0.081Steering-Yaw angle (5 degree) 10 0.00035 0.00067 20 0.00036 0.00095

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(top)

(bottom)

Figure 7. (Top) Steering—Yaw control; (bottom) improved GA (mGA) versus standard GA (sGA) fitness functions.

5. Conclusions

In this study, genetic algorithms, during a short operation period, were utilized to optimize the parameters of an autonomous hovercraft controller. The proposed method achieved good performances in terms of response (fast), stability (high), error (low), and overshoot (none at all). In addition, the improved GA methodology, which was implemented by make some simple changes inside the standard GA’s process that neglects/eliminates the chromosome decode step, displayed better performances in convergence speed. Especially, the modified GA can update the error fitness function in a smaller number of generations. It is undeniable that the improved GA is valuable in the optimization processes, particularly in optimizing the controller parameters. In further research, the authors would like to enhance the efficiency of the modified GA by using another error criteria, such as: ITSE (integral of time weighted square error) and MSE (mean square error). The disturbances attack to system models will be also considered to verify the efficiency of the process of optimizing control parameters.

Author Contributions: Methodology, H.K.T.; Supervision, H.K.T.; Visualization, H.K.T.; Resources, P.V.D., H.H.S.; Validation, T.T.T.; Writing-Review & Editing, H.-N.N. All authors have read and agree to the published version of the manuscript.

Figure 7. (Top) Steering—Yaw control; (bottom) improved GA (mGA) versus standard GA (sGA)fitness functions.

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5. Conclusions

In this study, genetic algorithms, during a short operation period, were utilized to optimize theparameters of an autonomous hovercraft controller. The proposed method achieved good performancesin terms of response (fast), stability (high), error (low), and overshoot (none at all). In addition, theimproved GA methodology, which was implemented by make some simple changes inside the standardGA’s process that neglects/eliminates the chromosome decode step, displayed better performancesin convergence speed. Especially, the modified GA can update the error fitness function in a smallernumber of generations. It is undeniable that the improved GA is valuable in the optimization processes,particularly in optimizing the controller parameters. In further research, the authors would like toenhance the efficiency of the modified GA by using another error criteria, such as: ITSE (integral oftime weighted square error) and MSE (mean square error). The disturbances attack to system modelswill be also considered to verify the efficiency of the process of optimizing control parameters.

Author Contributions: Methodology, H.K.T.; Supervision, H.K.T.; Visualization, H.K.T.; Resources, P.V.D., H.H.S.;Validation, T.T.T.; Writing—Review & Editing, H.-N.N. All authors have read and agreed to the published versionof the manuscript.

Funding: This work was financially supported by the “Center for Cyber-physical System Innovation” from TheFeatured Areas Research Center Program within the framework of the Higher Education Sprout Project by theMinistry of Education (MOE) in Taiwan.

Conflicts of Interest: The authors declare that there is no conflict of interests regarding the publication of this paper.

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